A Study on Asymptotic Expansion of the Risk-Neutral Pricing Density
A rich selection of liquidly traded vanilla and exotic contracts are used in modern nancial markets, depending on a large number of underlying ones. The clear valuation of novel and current derivative contracts to rule out arbitrage opportunities is a key requirement in such dense markets. The implementation of a new approach for pricing contingent claims is based on an asymptotic extension of the dynamics of pricing density. In a chosen coordinate frame, in which the price density looks stationary, the expansion is carried out. By using a complete set of orthogonal Hermite-polynomials, the resulting asymptotic Kolmogorov-backwards-equation is approximated. The derived model is calibrated and evaluated on a range of 1075 ‘Deutscher Aktienindex’ (DAX) index options in European style and is shown to deliver very precise option prices and a surface of implied volatility that is more precise than traditional methods. With all the advantages and disadvantages, even in competitive markets with exceptional circumstances, the proposed approach is very promising and well-suited for option pricing.
Department of Economics and Finance, University of Greifswald, 17489 Greifswald, Germany.
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